Hamiltonian Gromov-Witten theory(see Mundet-Tian paper) corresponds to a new type of Symplectic vortex equations: Such type of models gives a connection to Hitchin-Kobayashi correspondence and Floer theory(for example Donaldson's book: Floer homology groups in Yang–Mills theory).

Question:

Take the vortex equations with values in a symplectic manifold $(M, \omega)$. We assume that $(M, \omega)$ is equipped with a Hamiltonian action by a compact Lie group $G$ that is generated by an equivariant moment map $\mu : M \to \mathfrak g$. The symplectic vortex equations have the form $$\overline ∂_{J,A}(u) = 0, \; \; ∗F_A + \mu(u) = \tau\in \mathcal Z(\mathfrak g)$$ Here $ P\to Σ$ is a principal $G$-bundle over a compact Riemann surface, $u : P → M $ is an equivariant smooth function, and $A$ is a connection on $P$.

Now how can we extend the study of symplectic vortex equations to logarithmic setting. In fact if we assume $M$ is a log-symplectic manifold, then Gualtieri- Li-Pelayo and Ratiu developed log symplectic reduction and log moment map. So the study of Log symplectic vortex equations in Hamiltonian Log GW theory or its Hitchin-Kobayashi type correspondencis natural to be considered. Is there any progress about it. Any reference or comment is welcomed